Fontaine’s Forgotten Method for Inexact Differential Equations

Fontaine’s Forgotten Method for Inexact Differential Equations
Author(s): Aaron E. Leanhardt and Adam E. Parker

Mathematics Magazine, Vol. 90, No. 3 (June 2017), pp. 208-219.

Published by: Mathematical Association of America

Stable URL: http://www.jstor.org/stable/10.4169/math.mag.90.3.208

Tag: First order equations

Summary: One learns about exact and inexact differential equations in any ODE course. Depending on the emphasis of the course, it is often the case that a minimal amount of time will be spent on this topic. However, the authors point out that much can be learned if one views exact and inexact differential equations in an historical perspective. In 1739 Alexis-Claude Clairaut published the modern integrating factor method of solving inexact ordinary differential equations (ODEs). He was motivated by a 1738 Alexis Fontaine paper with a different method which requires solving a difficult partial differential equation (PDE). The authors revisit Fontaine's method, examine his modest attempt to solve the PDE, and utilize a different technique to give the first family of ODEs solvable by Fontaine's method with no obvious solution using the modern technique.

Review by Thomas W. Judson, Stephen F. Austin State University, June 7, 2017.

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